Monte Carlo Methods Ensembles (Chapter 5) Biased Sampling (Chapter - PowerPoint PPT Presentation

Monte Carlo Methods Ensembles (Chapter 5) Biased Sampling (Chapter 14) Practical Aspects

Lecture 1 2

Lecture 1&2 3

Lecture 1&3 4

Different Ensembles Ensemble Name Constant Fluctuating (Imposed) (Measured) NVT Canonical N,V,T P NPT Isobaric-isothermal N,P,T V µ VT Grand-canonical µ ,V,T N 5

NVT Liquids Equation of State of Liquid Carbon 6 Imposed

… if force is difficult to calculate … NPT e.g. carbon force field Liquids Equation of State of Liquid Carbon Imposed 7

NPT Non-Isotropic Systems e.g. Solids Structure and Transformation of Carbon Nanotube Arrays 8

µ VT Adsorption Adsorption in Carbon Nanostructues 9 Schwegler et al.

Lecture 2 Statistical Thermodynamics Partition function 1 [ ] ( ) N N Q dr exp U r = ∫ − β NVT 3 N N ! Λ Ensemble average 1 1 [ ] ( ) ( ) N N N A dr A r exp U r = ∫ − β 3 N NVT Q N ! Λ NVT Probability to find a particular configuration 1 1 ( N ) N ( N N ) ( N ) ( N ) N r dr' r' r exp U r' exp U r ⎡ ⎤ ⎡ ⎤ = ∫ δ − − β ∝ − β 3 N ⎣ ⎦ ⎣ ⎦ Q N ! Λ NVT Free energy ( ) F ln Q β = − 10 NVT

Lecture 2 Ensemble average 1 1 [ ] ( ) ( ) N N N A dr A r exp U r = ∫ − β 3 N NVT Q N ! Λ NVT ( ) ( ) N N N dr A r P r ∫ ( ) ( ) N N N dr A r P r = ∫ = ( ) N N dr P r ∫ [ ] ( ) ( ) ( ) [ ( ) ] N N N N N N dr A r C exp U r dr A r exp U r ∫ − β − β ∫ = = [ ] [ ] ( ) ( ) N N N N dr C exp U r dr exp U r ∫ − β ∫ − β Generate configuration using MC: ( ) ( ) N N MC N dr A r P r ∫ 1 M ( ) { } N A A r N N N N N ∑ ! r , r , r , r , r = = i ( ) M N MC N 1 2 3 4 M dr P r ∫ i 1 = [ ] ( ) ( ) N N MC N dr A r C exp U r Weighted Distribution ∫ − β with = [ ] ( ) N MC N dr C exp U r ∫ − β [ ] ( ) ( ) MC N MC N P r C exp U r [ ] = − β ( ) ( ) N N N dr A r exp U r ∫ − β = [ ] ( ) N N dr exp U r ∫ − β

Lecture 2 Monte Carlo: Detailed balance o n K o ( n ) K n ( o ) → = → K o ( n ) N o ( ) ( o n ) acc( o n ) → = × α → × → K n ( o ) N n ( ) ( n o ) acc( n o ) → = × α → × → acc( n → o ) = N ( n ) × α ( n → o ) acc( o → n ) N ( o ) × α ( o → n ) = N ( n ) N ( o )

Lecture 2 NVT -ensemble N n ( ) exp U n ( ) ⎡ ⎤ ∝ − β ⎣ ⎦ acc( o n ) N n ( ) → = acc( n o ) N o ( ) → acc( o n ) → exp U n U o ( ) ( ) ⎡ ⎤ ⎡ ⎤ = − β − ⎣ ⎦ ⎣ ⎦ acc( n o ) → 13

NPT ensemble We control the • Temperature (T) • Pressure (P) • Number of particles (N) 14

Intermezzo Scaled coordinates Partition function 1 [ ] ( ) N N Q dr exp U r = ∫ − β NVT 3 N N ! Λ Scaled coordinates s r i L / = The energy depends on i the real coordinates This gives for the partition function 3 N L N ( N ) Q ds exp U s ; L ⎡ ⎤ = − β ∫ NVT 3 N ⎣ ⎦ N ! Λ N V N ( N ) ds exp U s ; L ⎡ ⎤ = − β ∫ 3 N ⎣ ⎦ N ! Λ 15

The NPT ensemble Here they are V 0 : total volume an ideal gas M : total number of particles N in volume V M-N in volume V 0 -V Here they interact V 0 is fixed V varies from 0 to V 0 What is the statistical thermodynamics of this ensemble?

Fixed V The NPT ensemble: partition function N V ( ) N N Q ds exp U s ; L ⎡ ⎤ = ∫ − β NVT ⎣ ⎦ 3 N N ! Λ M − N ( ) V 0 − V V N ! # ( ) ds M − N exp − β U 0 s M − N ; L Q MV 0 , NV , T = ∫ Λ 3 N N ! Λ 3( M − N ) M − N " $ ( ) ! ! # ( ) ds N exp − β U s N ; L ∫ × " $ M − N ( ) V 0 − V V N $ & ( ) ds N exp − β U s N ; L Q MV 0 , NV , T = ∫ Λ 3( M − N ) M − N Λ 3 N N ! % ' ( ) ! 17

M − N ( ) V 0 − V V N $ & ( ) ds N exp − β U s N ; L Q MV 0 , NV , T = ∫ Λ 3( M − N ) M − N Λ 3 N N ! % ' ( ) ! To get the Partition Function of this system, we have to integrate over all possible volumes : M − N ( ) V 0 − V V N $ & ( ) ds N exp − β U s N ; L Q MV 0 , N , T = d V ∫ ∫ Λ 3( M − N ) M − N Λ 3 N N ! % ' ( ) ! Now let us take the following limits: M M → ∞⎫ constant ρ = → ⎬ V V → ∞ ⎭ 0 As the particles are an ideal gas in the big reservoir we have: P ρ = β 18

M − N ( ) V 0 − V V N $ & ( ) ds N exp − β U s N ; L Q MV 0 , N , T = d V ∫ ∫ Λ 3 N N ! Λ 3( M − N ) M − N % ' ( ) ! We have M N M N − − M N M N V V V 1 V V V exp M N V V ( ) − ( ) − ( ) ⎡ ⎤ − = − ≈ − − ⎣ ⎦ 0 0 0 0 0 M N − M N M N V V V − exp [ V ] V − exp [ PV ] ( ) − ≈ − ρ = − β 0 0 0 To make the partition function dimensionless This gives: (see Frenkel/Smit for more info) P β N N ( N ) Q d V exp PV V ds exp U s ; L [ ] ⎡ ⎤ = Λ ∫ − β ∫ − β NPT ⎣ ⎦ 3 N N ! 19

NPT Ensemble Partition function: P β N N ( N ) Q d V exp PV V ds exp U s ; L [ ] ⎡ ⎤ = Λ ∫ − β ∫ − β NPT ⎣ ⎦ 3 N N ! Probability to find a particular configuration: ( N ) N ( N ) N V , s V exp PV exp U s ; L [ ] ⎡ ⎤ ∝ − β − β NPT ⎣ ⎦ De Detailed balance Sample a particular configuration: • change of volume • change of reduced coordinates Acceptance rules ?? 20

Detailed balance o n K o ( n ) K n ( o ) → = → K o ( n ) N o ( ) ( o n ) acc( o n ) → = × α → × → K n ( o ) N n ( ) ( n o ) acc( n o ) → = × α → × → acc( o n ) N n ( ) ( n o ) N n ( ) → × α → = = acc( n o ) N o ( ) ( o n ) N o ( ) → × α → 21

NPT -ensemble ( N ) N ( N ) N V , s V exp PV exp U s ; L [ ] ⎡ ⎤ ∝ − β − β NPT ⎣ ⎦ acc( o n ) N n ( ) → = acc( n o ) N o ( ) → Suppose we change the position of a randomly selected particle N ( N ) V exp [ PV ] exp U s ; L ⎡ ⎤ − β − β acc( o n ) → n ⎣ ⎦ = acc( n o ) N ( N ) V exp PV exp U s ; L → [ ] ⎡ ⎤ − β − β o ⎣ ⎦ ( N ) exp U s ; L ⎡ ⎤ − β n ⎣ ⎦ { } exp U n U o ( ) ( ) ⎡ ⎤ = = − β − ⎣ ⎦ ( N ) exp U s ; L ⎡ ⎤ − β o ⎣ ⎦ 22

NPT -ensemble ( N ) N ( N ) N V , s V exp PV exp U s ; L [ ] ⎡ ⎤ ∝ − β − β NPT ⎣ ⎦ acc( o n ) N n ( ) → = acc( n o ) N o ( ) → Suppose we change the volume of the system N ( N ) V exp [ PV ] exp U s ; L ⎡ ⎤ − β − β acc( o n ) → n n n ⎣ ⎦ = acc( n o ) N ( N ) V exp PV exp U s ; L → [ ] ⎡ ⎤ − β − β o o o ⎣ ⎦ N V ⎛ ⎞ { } n exp P V V exp U n U 0 ( ) ( ) ( ) = ⎡ − β − ⎤ − β ⎡ − ⎤ ⎜ ⎟ ⎣ ⎦ ⎣ ⎦ n o V ⎝ ⎠ o 23

Algorithm: NPT • Randomly change the position of a particle • Randomly change the volume 24

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NPT simulations Equation of State of Lennard Jones System 28

Measured and Imposed Pressure • Imposed pressure P • Measured pressure <P> from virial # & ds N e − β U ( s N ) ∂ F dVV N e − β PV ∫ ∫ − % ( # & ∂ V P = − < ∂ F $ ' N , T ( > N , T = % ds N e − β U ( s N ) dVV N e − β PV $ ∂ V ' ∫ ∫ * , ( ) p ( V ) = exp − β F ( V ) + PV + - Q NPT ∫ * , ( ) Q NPT = β P dV exp − β F ( V ) + PV + - 29

# & ds N e − β U ( s N ) ∂ F ∫ dVV N e − β PV ∫ − % ( # & P = − < ∂ F $ ∂ V ' N , T ( > N , T = % ds N e − β U ( s N ) dVV N e − β PV $ ∂ V ' ∫ ∫ # & β P dV ∂ F * , ( ) ∫ ( ) + PV P = − exp − β F V % ( + - Q ( NPT ) $ ∂ V ' N , T * , ( ) ∂ exp − β F V [ ] dV exp − β PV β P + - ∫ P = Q ( NPT ) ∂ V β 30

Measured and Imposed Pressure b b b • Partial integration [ ] fdg fg gdf ∫ = − ∫ a a a " $ ( ) ( ) + PV exp − β F V % = 0 • For V=0 and V= ∞ # • Therefore, ! # ( ) ∂ exp − β F V [ ] dV exp − β PV β P " $ ∫ P = Q ( NPT ) ∂ V β β P ! # ( ) ∫ ( ) + PV P = dVP exp − β F V = P Q ( NPT ) = " $ 31

Grand-canonical ensemble What are the equilibrium conditions? 32

Grand-canonical ensemble We impose: – Temperature (T) – Chemical potential ( µ ) – Volume (V) – But NOT pressure 33

The ensemble of the total system Here they are an ideal gas Here they interact What is the statistical thermodynamics of this ensemble? 34